Question

The mass flow rate at the inlet of a one-inlet, one-exit control volume varies with time according to \(\dot{m}_{\mathrm{i}}=100\left(1-e^{-2 l}\right)\), where \(\dot{m}_{1}\) has units of \(\mathrm{kg} / \mathrm{h}\) and \(t\) is in \(\mathrm{h}\). At the exit, the mass flow rate is constant at \(100 \mathrm{~kg} / \mathrm{h}\). The initial mass in the control volume is \(50 \mathrm{~kg}\). (a) Plot the inlet and exit mass flow rates, the instantaneous rate of change of mass, and the amount of mass contained in the control volume as functions of time, for \(t\) ranging from 0 to \(3 \mathrm{~h}\). (b) Estimate the time, in \(\mathrm{h}\), when the tank is nearly empty.

Short answer

Plot the inlet and exit mass flow rates, the instantaneous rate of change of mass, and the mass in the control volume. The tank is nearly empty when \( m(t) \) approaches 0, which can be determined from the plot.

Step-by-step solution

- Understand the Given Functions

The inlet mass flow rate is given by \ \( \dot{m}_i = 100(1 - e^{-2t}) \) kg/h, and the exit mass flow rate is constant at \ \( \dot{m}_e = 100 \) kg/h. The initial mass in the control volume is 50 kg. We want to plot the inlet and exit mass flow rates as well as the instantaneous rate of change of mass and the amount of mass in the control volume as a function of time for \( t \) ranging from 0 to 3 h.

- Plot the Inlet Mass Flow Rate

To plot \( \dot{m}_i \), create a time vector from 0 to 3 hours. Calculate \( \dot{m}_i \) at each time point using the given function \( \dot{m}_i = 100(1 - e^{-2t}) \). Plot the resulting values against time.

- Plot the Exit Mass Flow Rate

The exit mass flow rate \( \dot{m}_e \) is constant at 100 kg/h. Plot a horizontal line at 100 kg/h from \( t = 0 \) to \( t = 3 \) hours to represent \( \dot{m}_e \).

- Calculate and Plot the Instantaneous Rate of Change of Mass

The rate of change of mass within the control volume can be calculated as \( \frac{dm}{dt} = \dot{m}_i - \dot{m}_e \). Using the functions for \( \dot{m}_i \) and \( \dot{m}_e \), this can be written as \( \frac{dm}{dt} = 100(1 - e^{-2t}) - 100 \). Evaluate and plot this function over the range of 0 to 3 hours.

- Calculate and Plot the Mass in the Control Volume

To calculate the mass in the control volume \( m(t) \), integrate the rate of change of mass \( \frac{dm}{dt} \) with respect to time. Use the initial condition \( m(0) = 50 \) kg. Set up the integral \( m(t) = \int_0^t (100(1 - e^{-2\tau}) - 100) d\tau + 50 \) and evaluate numerically for \( t \) from 0 to 3 hours. Plot \( m(t) \) against time.

- Estimate the Time When the Tank is Nearly Empty

The tank is considered to be nearly empty when the mass \( m(t) \) is close to 0. From the plot of \( m(t) \), find the time at which \( m(t) \) approaches zero and interpret this time as when the tank is nearly empty.

Key concepts

Mass Conservation

Mass conservation is a fundamental principle in fluid mechanics and thermodynamics. It states that mass cannot be created or destroyed within a closed system, only transferred in and out. In the context of this exercise, we have a control volume with one inlet and one exit. The mass flow rates at both points must be analyzed to ensure mass balance. We are given the inlet mass flow rate \( \dot{m}_i = 100(1 - e^{-2t}) \ kg/h \), which changes over time and the exit mass flow rate \( \dot{m}_e \) that remains constant at 100 kg/h. The goal is to track how mass is conserved in the system over time, ensuring the principle of mass conservation is upheld.

Control Volume

A control volume is an imaginary boundary around the system where mass flow is studied. This boundary can be fixed or moving but helps in applying the principles of fluid mechanics. For this exercise, our control volume is the tank that initially contains 50 kg of mass. Over time, the inlet mass flow rate and exit mass flow rate affect the total mass within the control volume. To analyze this, we need to calculate mass at multiple time intervals using the given functions \( \dot{m}_i = 100(1 - e^{-2t}) kg/h \) and \( \dot{m}_e = 100 kg/h\).

Rate of Change

The rate of change principle helps us understand how quickly a quantity is increasing or decreasing over time. In this exercise, the rate of change of mass within the control volume can be expressed as \( \frac{dm}{dt} = \dot{m}_i - \dot{m}_e\). This formula allows us to calculate how mass changes at any given moment. We use the provided mass flow rate equations to determine this rate from 0 to 3 hours. As time progresses, the inlet mass flow rate begins high and eventually stabilizes, while the exit mass flow rate remains unchanged. Integrating this rate of change over time tells us the total mass remaining at any given point, helping us estimate when the tank will be nearly empty.